Finite trees as ordinals Herman Ruge Jervell University of Oslo - PowerPoint PPT Presentation

Finite trees as ordinals Herman Ruge Jervell University of Oslo Honouring Wilfried M¨ unchen April 5, 2008

Typical trees The natural numbers: 0 = ·

Typical trees The natural numbers: · 0 = · 1 = ·

Typical trees The natural numbers: · · · 0 = · 1 = · 2 = ·

Typical trees The natural numbers: · · · · · · 0 = · 1 = · 2 = · 3 = ·

Typical trees The natural numbers: · · · · · · 0 = · 1 = · 2 = · 3 = · And some ordinals:

Typical trees The natural numbers: · · · · · · 0 = · 1 = · 2 = · 3 = · And some ordinals: · · ω = ·

Typical trees The natural numbers: · · · · · · 0 = · 1 = · 2 = · 3 = · And some ordinals: · · · · · ω ω = ω = · ·

Typical trees The natural numbers: · · · · · · 0 = · 1 = · 2 = · 3 = · And some ordinals: · · · · · · · · ω ω = ω = ǫ 0 = · · ·

Typical trees The natural numbers: · · · · · · 0 = · 1 = · 2 = · 3 = · And some ordinals: · · · · · · · · · · · · ω ω = ω = ǫ 0 = Γ 0 = · · · ·

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B �

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B �

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B all immediate subtrees of A are < B

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B all immediate subtrees of A are < B ◮ � A � < � B �

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B all immediate subtrees of A are < B ◮ � A � < � B �

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B all immediate subtrees of A are < B ◮ � A � < � B � lexicographical ordering

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B all immediate subtrees of A are < B ◮ � A � < � B � lexicographical ordering ◮ Length of sequences

The ordering A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ � A � — sequence of immediate subtrees ◮ A ≤ � B � A ≤ some immediate subtree of B ◮ � A � < B all immediate subtrees of A are < B ◮ � A � < � B � lexicographical ordering ◮ Length of sequences ◮ Rightmost element where they differ

Elementary properties A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ Decidable

Elementary properties A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ Decidable ◮ Transitive

Elementary properties A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ Decidable ◮ Transitive ◮ Linear

Elementary properties A < B ⇔ A ≤ � B � ∨ ( � A � < B ∧ � A � < � B � ) ◮ Decidable ◮ Transitive ◮ Linear ◮ Equality is the usual tree equality

Some ordinal functions Zero: · = 0

Some ordinal functions Zero: · = 0 Successor: α = α + 1 ·

Some ordinal functions Zero: · = 0 Successor: α = α + 1 · Exponentiation: · α ∼ ω ω α · where ∼ means we jump over fix points.

Some ordinal functions Zero: · = 0 Successor: α = α + 1 · Exponentiation: · α ∼ ω ω α · where ∼ means we jump over fix points. In general we get the fix point free n -ary Veblen functions.

Approximating from below 1 · · · · Γ 0 = · Start with immediate subtrees:

Approximating from below 1 · · · · Γ 0 = · Start with immediate subtrees: · 0 = · 0 = · 1 = ·

Approximating from below 1 · · · · Γ 0 = · Start with immediate subtrees: · 0 = · 0 = · 1 = · Use function with smaller arity:

Approximating from below 1 · · · · Γ 0 = · Start with immediate subtrees: · 0 = · 0 = · 1 = · Use function with smaller arity: γ α β · ·

Approximating from below 1 · · · · Γ 0 = · Start with immediate subtrees: · 0 = · 0 = · 1 = · Use function with smaller arity: γ α β · ·

Approximating from below 2 · · · · Γ 0 = · Less in lexicographical ordering:

Approximating from below 2 · · · · Γ 0 = · Less in lexicographical ordering: · α β ·

Approximating from below 2 · · · · Γ 0 = · Less in lexicographical ordering: · α β · This gives all trees less than Γ 0 .

Approximating from below 2 · · · · Γ 0 = · Less in lexicographical ordering: · α β · This gives all trees less than Γ 0 . To get a cofinal set we only need γ · · ·

Wellfoundedness ◮ Minimal bad argument

Wellfoundedness ◮ Minimal bad argument ◮ Minimal height

Wellfoundedness ◮ Minimal bad argument ◮ Minimal height ◮ Induction over wellfounded trees

Wellfoundedness ◮ Minimal bad argument ◮ Minimal height ◮ Induction over wellfounded trees

Wellfoundedness ◮ Minimal bad argument ◮ Minimal height ◮ Induction over wellfounded trees Both arguments are straightforward.

Further work Linear extensions of embeddings ◮ Diana Schmidt

Further work Linear extensions of embeddings ◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees

Further work Linear extensions of embeddings ◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees ◮ | A | maximal ordertype

Further work Linear extensions of embeddings ◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees ◮ | A | maximal ordertype

Further work Linear extensions of embeddings ◮ Diana Schmidt ◮ Linear extensions of topological embeddings of trees ◮ | A | maximal ordertype | A | | B | | B | | A | � B � A � � � ≤ ⊕ � � · · · � � � This gives Higmans lemma. Further work gives Kruskals theorem.

Further work Finite trees with labels ◮ Wellordered set of labels

Further work Finite trees with labels ◮ Wellordered set of labels ◮ Each node has a label

Further work Finite trees with labels ◮ Wellordered set of labels ◮ Each node has a label ◮ � A � i – sequence of i -subtrees

Further work Finite trees with labels ◮ Wellordered set of labels ◮ Each node has a label ◮ � A � i – sequence of i -subtrees ◮ Defines < i and < ∞

Further work Finite trees with labels ◮ Wellordered set of labels ◮ Each node has a label ◮ � A � i – sequence of i -subtrees ◮ Defines < i and < ∞ ◮ A < i B ⇔ A ≤ i � B � i ∨ ( � A � i < B ∨ A < i + B )

Further work Finite trees with labels ◮ Wellordered set of labels ◮ Each node has a label ◮ � A � i – sequence of i -subtrees ◮ Defines < i and < ∞ ◮ A < i B ⇔ A ≤ i � B � i ∨ ( � A � i < B ∨ A < i + B ) ◮ A < ∞ B — lexicographical ordering

Further work Finite trees with labels ◮ Wellordered set of labels ◮ Each node has a label ◮ � A � i – sequence of i -subtrees ◮ Defines < i and < ∞ ◮ A < i B ⇔ A ≤ i � B � i ∨ ( � A � i < B ∨ A < i + B ) ◮ A < ∞ B — lexicographical ordering ◮ Linear wellfounded preorderings